Subgroups — #25065891

Order 11008.

616 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143721	C₂	1	2	yes	maximal
2	C₈×#10621898	C₂	1	1	yes	maximal, characteristic
3	#19133386	C₂	1	2	yes	maximal, complemented
4	#19133377	C₂	1	4	yes	maximal, complemented
5	#19133355	C₂	1	2	yes	maximal, complemented
6	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×D₄₃	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10621898	C₂²	1	1	yes	characteristic
10	#14583026	—	2	2	no	—
11	C₈×Dic₈₆	C₄	1	1	yes	characteristic
12	C₈×Dic₈₆	C₄	1	1	yes	characteristic
13	#14582968	—	2	4	no	—
14	C₈×D₁₇₂	C₄	1	1	yes	characteristic
15	C₈×D₁₇₂	C₄	1	1	yes	characteristic
16	#14582964	—	2	2	no	—
17	C₈×C₂²⋊D₄₃	C₄	1	2	yes	—
18	C₈×C₂²⋊D₄₃	C₄	1	2	yes	—
19	#14582955	C₂²	1	4	yes	—
20	#14582952	C₂²	1	2	yes	—
21	#14582952	C₂²	1	1	yes	characteristic
22	#14582950	C₂²	1	2	yes	—
23	C₄⋊C₈×D₄₃	C₂²	1	4	yes	—
24	C₄×C₈×D₄₃	C₂²	1	2	yes	—
25	#14582652	C₂²	1	2	yes	—
26	#14582651	C₂²	1	4	yes	—
27	#14582650	C₂²	1	1	yes	characteristic
28	#14582650	C₂²	1	1	yes	characteristic
29	#14582649	C₂²	1	4	yes	—
30	C₄₃×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#334	C₂²	1	2	yes	—
33	C₄₃×#334	C₂²	1	2	yes	—
34	C₂×#10621898	Q₈	1	1	yes	characteristic
35	C₂×C₈×D₄₃	—	2	4	no	—
36	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
38	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₈₆	D₄	1	1	yes	characteristic
40	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
41	C₄×D₁₇₂	D₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
43	C₄×C₂²⋊D₄₃	D₄	1	2	yes	—
44	#13063284	Q₈	1	1	yes	characteristic
45	#13063282	Q₈	1	2	yes	—
46	C₄²×D₄₃	C₂³	1	2	yes	—
47	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
49	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
50	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
51	Dic₄₃⋊C₈	—	2	4	no	—
52	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
53	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
54	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
55	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
56	(C₂×D₄₃)⋊C₈	—	2	4	no	—
57	(C₂×D₄₃)⋊C₈	C₂×C₄	1	2	yes	—
58	(C₂×D₄₃)⋊C₈	C₂×C₄	1	2	yes	—
59	#13063235	—	2	4	no	—
60	C₄×C₄₃⋊C₈	—	2	2	no	—
61	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
62	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₄₃	—	2	2	no	—
65	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₃	—	2	2	no	—
69	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₂²⋊D₄₃	C₄⋊C₄	1	2	yes	—
78	#10621898	—	2	4	no	—
79	C₂×D₁₇₂	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
83	C₂×Dic₈₆	C₄⋊C₄	1	1	yes	characteristic
84	M₁₆×C₄₃	—	8	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	4	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
97	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	2	yes	—
98	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	2	yes	—
99	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
101	Dic₄₃⋊C₄	C₄⋊C₄	1	2	yes	—
102	Dic₄₃⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
104	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
105	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
110	C₈×D₄₃	—	2	4	no	—
111	C₈×D₄₃	—	4	2	no	—
112	C₈×D₄₃	—	8	4	no	—
113	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	2	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂²⋊D₄₃	—	2	4	no	—
126	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
129	D₁₇₂	#108	1	4	yes	—
130	C₄×D₄₃	—	2	4	no	—
131	C₄×D₄₃	—	2	2	no	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	4	no	—
134	Dic₈₆	—	2	2	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	4	4	no	—
139	C₈×C₄₃	#131	1	2	yes	—
140	C₄₃⋊C₈	—	2	2	no	—
141	C₄₃⋊C₈	—	2	2	no	—
142	C₄₃⋊C₈	—	4	2	no	—
143	C₄₃⋊C₈	—	4	2	no	—
144	#17834	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂×D₄₃	—	2	2	no	—
150	C₂×D₄₃	—	2	2	no	—
151	C₂×D₄₃	C₂×#108	1	4	yes	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	C₂×#108	1	2	yes	—
156	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
157	C₄×C₄₃	#568	1	1	yes	characteristic
158	C₄×C₄₃	#577	1	2	yes	—
159	Dic₄₃	—	2	2	no	—
160	Dic₄₃	—	2	2	no	—
161	Dic₄₃	—	2	2	no	—
162	#2990	—	43	2	no	—
163	C₈×#41	—	43	1	no	—
164	#1574	—	43	2	no	—
165	#1573	—	43	4	no	—
166	#1568	—	43	2	no	—
167	#1566	D₄₃	1	1	yes	characteristic, complemented
168	#1564	—	43	1	no	—
169	C₂×#334	—	43	2	no	—
170	C₂×C₄₃	—	4	1	no	—
171	C₂×C₄₃	#2906	1	1	yes	characteristic
172	C₂×C₄₃	#2912	1	1	yes	characteristic
173	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
174	D₄₃	—	2	4	no	—
175	C₄×#41	—	43	1	no	—
176	#446	—	86	2	no	—
177	C₈×Q₈	—	43	1	no	—
178	C₈×Q₈	—	43	1	no	—
179	#435	—	86	4	no	—
180	C₈×D₄	—	43	2	no	—
181	C₈×D₄	—	43	2	no	—
182	C₈×D₄	—	43	1	no	—
183	C₈×D₄	—	43	1	no	—
184	C₈×D₄	—	86	2	no	—
185	#432	—	43	4	no	—
186	#431	—	43	1	no	—
187	#431	—	43	2	no	—
188	#431	—	43	2	no	—
189	#423	C₂×D₄₃	1	2	yes	—
190	C₂×C₄⋊C₈	—	43	4	no	—
191	C₂×C₄×C₈	—	43	2	no	—
192	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
193	#335	—	43	4	no	—
194	#335	—	43	2	no	—
195	#334	—	43	4	no	—
196	#334	—	43	1	no	—
197	#334	—	43	1	no	—
198	#334	C₂×D₄₃	1	2	yes	—
199	#334	C₂×D₄₃	1	2	yes	—
200	C₄₃	#17834	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	43	1	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂²×C₈	—	43	2	no	—
204	C₂²×C₈	—	43	2	no	—
205	C₂²×C₈	—	86	4	no	—
206	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
208	C₄×Q₈	—	43	1	no	—
209	C₄×Q₈	—	43	1	no	—
210	C₄×D₄	—	43	2	no	—
211	C₄×D₄	—	43	2	no	—
212	C₄×D₄	—	43	1	no	—
213	C₄×D₄	—	43	1	no	—
214	#117	—	43	1	no	—
215	#117	—	43	2	no	—
216	C₂×C₄²	—	43	2	no	—
217	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	43	2	no	—
220	C₄⋊C₈	—	43	2	no	—
221	C₄⋊C₈	—	43	1	no	—
222	C₄⋊C₈	—	43	1	no	—
223	C₄⋊C₈	—	43	2	no	—
224	C₄⋊C₈	—	43	2	no	—
225	C₄⋊C₈	—	86	4	no	—
226	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
227	C₂²⋊C₈	—	43	2	no	—
228	C₂²⋊C₈	—	43	2	no	—
229	C₂²⋊C₈	—	43	2	no	—
230	C₂²⋊C₈	—	86	4	no	—
231	C₂²⋊C₈	—	86	2	no	—
232	#97	—	86	4	no	—
233	C₄×C₈	—	43	2	no	—
234	C₄×C₈	—	43	2	no	—
235	C₄×C₈	—	43	2	no	—
236	C₄×C₈	—	86	2	no	—
237	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
238	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	#41	—	86	4	no	—
242	C₂×Q₈	—	43	1	no	—
243	C₂×D₄	—	43	2	no	—
244	C₂×D₄	—	43	1	no	—
245	C₂²×C₄	—	43	2	no	—
246	C₂²×C₄	—	43	2	no	—
247	C₂²×C₄	—	43	2	no	—
248	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
250	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
251	M₁₆	—	8	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	43	2	no	—
256	C₂×C₈	—	43	2	no	—
257	C₂×C₈	—	86	4	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	86	2	no	—
260	C₂×C₈	—	344	4	no	—
261	C₂×C₈	—	172	2	no	—
262	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₄⋊C₄	—	43	2	no	—
267	C₄⋊C₄	—	43	2	no	—
268	C₄⋊C₄	—	43	1	no	—
269	C₄⋊C₄	—	43	1	no	—
270	C₂²⋊C₄	—	43	2	no	—
271	C₂²⋊C₄	—	43	2	no	—
272	C₂²⋊C₄	—	43	2	no	—
273	C₄²	—	43	2	no	—
274	C₄²	—	43	2	no	—
275	C₄²	—	43	2	no	—
276	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
277	C₄²	D₄×D₄₃	1	1	yes	characteristic
278	C₄²	Q₈×D₄₃	1	1	yes	characteristic
279	C₄²	C₂³×D₄₃	1	1	yes	characteristic
280	C₂³	—	43	2	no	—
281	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
282	Q₈	—	86	2	no	—
283	D₄	—	43	4	no	—
284	D₄	—	86	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	43	2	no	—
290	C₂×C₄	—	43	2	no	—
291	C₂×C₄	—	43	2	no	—
292	C₂×C₄	—	86	4	no	—
293	C₂×C₄	—	86	2	no	—
294	C₂×C₄	—	86	2	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	4	no	—
306	C₈	—	86	2	no	—
307	C₈	—	86	2	no	—
308	C₈	—	172	2	no	—
309	C₈	—	172	2	no	—
310	C₈	#13063333	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	43	4	no	—
315	C₂²	—	86	2	no	—
316	C₂²	—	86	2	no	—
317	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	86	2	no	—
322	C₄	—	86	2	no	—
323	C₄	—	86	2	no	—
324	C₄	#108×D₄₃	1	2	yes	—
325	C₄	C₂×#13063333	1	1	yes	characteristic
326	C₄	#14583665	1	1	yes	characteristic
327	C₄	#14583705	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	86	4	no	—
330	C₂	#19143164	1	1	yes	characteristic
331	C₂	#19143186	1	1	yes	characteristic
332	C₂	C₂×#108×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1566	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂×#10621898	Q₈
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1566	D₄₃
hypocenter	C₄₃	#17834
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #19133386	C₂	semidirect
4. #19133377	C₂	semidirect
5. #19133355	C₂	semidirect
6. #19133343	C₂	semidirect
7. #334×D₄₃	C₂	semidirect
8. C₄₃×#1566	C₂	semidirect
167. #1566	D₄₃	semidirect
200. C₄₃	#17834	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19143721	2
C₈×#10621898	2
#19133386	2
#19133377	2
#19133355	2
#19133343	2
#334×D₄₃	2
C₄₃×#1566	2
#17834	43